Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension. (Lean)
Manuscript
A zero-entropy system without a smooth positive-volume model
September 25, 2026 · source · cite as \cite{OAI:A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026}
We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.
Result 152 · Dynamical systems and ergodic theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md